It seems fairly well known that Leray originated the ideas of spectral sequences and sheaves while being held in a prisoner of war camp in Austria from 1940 to 1945. Weil famously proved the Riemann hypothesis for curves in 1940, while in prison for failure to report for army duty. I recently learned that Linnik's famous theorem on primes in arithmetic progressions was published in 1944, just after the siege of Leningrad ended. So now I would like to ask:

What are some other examples of notable mathematics done during World War II?

My recollection is that Turan's work obtained in forced labor camp (not concentration camp) was on the crossing number on complete bipartite graphs. His description of this is quoted on p 50 of "Geometric graphs and arrangements: some chapters from Combinational geometry" by Stefan Felsner available on books.google.com. Worth reading!
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Péter KomjáthNov 24 '10 at 6:56

26 Answers
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On the other side of the war, Teichmüller did some of his best work during World War II. According to the MacTutor biography, he volunteered to serve on the Eastern Front in 1943 and got killed. My impression, then, is that his Nazi fanaticism was a crime against his own mathematical career as well as against other mathematicians.

I remember reading a interesting article from the AMS a while ago about the Japanese mathematician Mikio Sato, who independently did some important work in algebraic analysis during the World War II. If my memory serves me well he was developing his theory of hyperfunctions at a young age all the while having to feed and protect his family during the war and "carrying coal" to earn a living. Here is a link to the AMS article: http://www.ams.org/notices/200702/fea-sato-2.pdf

Edit: Since it hasn't yet been mentioned, Alan Turing did great work during WW-II: he participated in a team that cracked the Enigma machine and many other codes/cyphers. http://en.wikipedia.org/wiki/Alan_Turing

The story of Wolfgang Doeblin. Results remained unknown till 2000.
See "Comments on the life and mathematical legacy of Wolfgang Doeblin",
by Bernard Bru and Marc Yor (link)
There is also a documentary.

According to Wikipedia: "During World War II [Selberg] worked in isolation due to the German occupation of Norway. After the war his accomplishments became known, including a proof that a positive proportion of the zeros of the Riemann zeta function lie on the [critical] line"
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Micah MilinovichAug 19 '10 at 17:20

Eilenberg and Mac Lane's papers on category theory started appearing: "Natural Isomorphisms in Group Theory" in the Proc. National Acad. Sci. USA in 1942 and "General Theory of Natural Equivalences" in Transactions of the AMS in 1945.

That doesn't quite fit David's request for work done in wartime conditions. Mathematicians in the US were not exactly under siege! A more suitable example would be the Gelfand--Naimark theorem characterizing C*-algebras and the Gelfand--Raikov theorem showing that the points in any locally compact group can be separated by some irreducible unitary representation of the group. These both appeared in 1943.

G. Hochschild's "On the Cohomology Groups of an Associative Algebra" (Annals of Mathematics 1945, the paper that founded Hochschild cohomology) addresses the author at "Aberdeen Proving Ground, Md.". It seems to me that this is by far not the only case of mathematicians working for the US military during and directly after WWII (although the Manhattan Project staff would hardly have put their locations on their publications).
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darij grinbergAug 19 '10 at 20:03

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At a conference 11 or 12 years ago, Saunders Mac Lane spoke briefly about -- as I recall -- he and Eilenberg "working on codes during the day and on cohomology of groups at night" during the war.
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Jeff StromAug 21 '10 at 13:40

George Dantzig essentially developed the foundations of linear programming while he was under the employment of the military. As has been mentioned in books, the term "programming" itself in this context is military terminology. (The simplex method however came after the war, in 1947).

was not only written during the war, but also was stimulated by the war. Subsequently it played an important role in prehistory of hyperbolic dynamics.

In 1960 Stephen Smale conjectured that Morse-Smale systems are the only structurally
stable systems.
It was pointed out to Smale that his conjectures are likely to be false. Rene
Thom argued that hyperbolic automorphism does not lie in the closure of Morse-
Smale systems. Norman Levinson wrote to Smale with a reference to the above paper in
which Cartwright and Littlewood studied certain differential equation of second
order with periodic forcing. This work arose from war-related studies involving
radio waves. The equation leads to a flow on R3. According to Levinson this
flow
has infinitely many periodic orbits; this phenomenon is robust which can be seen
from the paper and also it was directly proved for a dierent equation in his own
work. This led Smale to discovery of the famous horseshoe and subsequent explosive development in smooth dynamics.

Gentzen published this paper in 1943 which initiated ordinal proof theory. I find it quite remarkable that he (Gentzen) could continue his logical studies after 1933, although Bieberbach obsessively tried to establish his 'German mathematics', a strange product of racism and misinterpreted intuitionism.

During the Second World War the theory of stochastic observation of a time-invariant process was developed by Wiener in the US and Kolmogorov in the USSR almost simultaneously. The results were published in a classified report which was declassified after the war, "Extrapolation, interpolation, and smoothing of stationary time series, with engineering applications".

Of course Switzerland was one of the few countries where mathematicians could basically do their business as usual, during WW2. Many fundamental discoveries of the Zurich school on algebraic topology (Hopf, Stiefel, Eckmann...) took place during this period. The journal Commentarii Mathematici Helvetici was published without interruption, and it is worth having a look at the Tables of contents (see e.g. http://retro.seals.ch/digbib/en/vollist?UID=comahe-001,comahe-002,comahe-003) to see that it was probably the best european journal during the wartime period.

To complete Tolland's answer, John von Neumann was the leading mathematician in Manhattan project. In this context, he started the mathematical analysis of multi-dimensional shock waves in the Euler equations of gas dynamics.

Grothendieck went to Vietnam to deliver lectures and a report of what he did can still be found online.

Bertrand Russell was imprisoned during WWI for anti-war activities and wrote "Introduction to Mathematical Philosophy" (1919) while in prison.

Hardy, in protest for Russell's affair, left Cambridge to Oxford and continued working there and collaborating by mail with Littlewood. Both of them worked during that time in Mathematics and there is fiction written about it.

Don't forget the cryptography work done by Turing, Welchman, and others during the war. The "Theorem that won World War II" (Rejewski's original group-theoretic attack on the Enigma encryption) was actually done shortly before the war, though.

I doubt it was Leray, since the French word for sheaf is faisceau which translates to "beam". I remember hearing a story that Norman Steenrod and someone else came up with the English words sheaf, stalk, and germ sitting on a front porch somewhere in the American midwest. I can't seem to find this story via google, but it seems possible since Steenrod was born in Dayton, Ohio. What do the French call stalks and germs?
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Jamie WeigandtAug 29 '12 at 20:06

The 1st Edition of Abraham Fraenkel’s book Einleitung in die Mengenlehre (Introduction to Set Theory) went to press during World War I. Fraenkel had teached set theory to his comrades while being at war, and this book were his lecture notes, so to say. He also gave his venia legendi lecture during the war while being on furlough.

Operations research was developed under WWII! This is mentioned in other answers, but only as "mathematical programming", while OR is much wider than that. One paper says

" Operations Research is a ‘war baby’. It is because, the first problem attempted to solve in a
systematic way was concerned with how to set the time fuse bomb to be dropped from an aircraft on
to a submarine. In fact the main origin of Operations Research was during the Second World War. "

googling for "operations research second world war" (or throw into that "submarine") gives a lot of information, one example which looks interesting is